Question

Calculate the resonant frequency and Q-factor (Quality factor) of a series L-C-R circuit containing a pure inductor of $$4\text{ H}$$, capacitor of capacitance $$27\mu \text{F}$$ and resistor of resistance $$8.4\mathrm{\Omega }$$.

Answer 1

Hint: At resonant frequency, the current in an L-C-R circuit will be maximum, which is possible only when inductive reactance $$X_L$$ is equal to capacitive reactance $$X_c$$.
At $$\omega ={\omega }_o$$, $$X_L=X_c$$, that is,
$${\omega }_{o}L={\displaystyle \frac{1}{{\omega }_{o}C}}$$

Formula used:
The resonant frequency $$f_o$$ of a series L-C-R is given by
$$f_o={\displaystyle \frac{1}{2\pi \sqrt{LC}}}$$
Here L denotes the inductance of the inductor and C denotes the capacitance of the capacitor.
The Q-factor of the circuit is s given by
$$Q={\displaystyle \frac{1}{R}}\sqrt{{\displaystyle \frac{L}{C}}}$$
Here L denotes the inductance of the inductor, C denotes the capacitance of the capacitor, and R denotes the resistance of the resistor.

Complete step by step solution:
Inductance, $$L=4\text{ H}$$
Capacitance, $$C=27\mu \text{F}$$
Resistance, $$R=8.4\mathrm{\Omega }$$
To find the resonant frequency, substitute the values of L and C in the resonant frequency formula:
$$\begin{array}{rl}& f_o={\displaystyle \frac{1}{2\pi \sqrt{(4\text{ H)(27 }\mu \text{F)}}}}\\ & f_o={\displaystyle \frac{1}{2\pi \sqrt{(4\text{ H)(27}\times \text{1}{\text{0}}^{-6}\text{F)}}}}\\ & f_o=15.31\text{ Hz}\end{array}$$
Now, to find the Q-factor of the circuit, substitute the values of L, C, and R in the Q-factor frequency formula:
$$\begin{array}{rl}& Q={\displaystyle \frac{1}{8.4\mathrm{\Omega }}}\sqrt{{\displaystyle \frac{4\text{ H}}{27\mu \text{F}}}}\\ & Q={\displaystyle \frac{1}{8.4\mathrm{\Omega }}}\sqrt{{\displaystyle \frac{4\text{ H}}{\text{27}\times \text{1}{\text{0}}^{-6}\text{F}}}}\\ & Q=45.82\end{array}$$

Therefore, the resonant frequency of the L-C-R circuit is $$15.31\text{ Hz}$$ and the Q-factor is $$45.82$$.

Additional information:
In general resonance is a phenomenon in which the natural frequency of a harmonic oscillator matches with the frequency of an external periodic force which in turn increases the amplitude of vibration.
Resonance is the result of oscillations in an L-C-R circuit as stored energy is passed from the inductor to the capacitor.
The sharpness of resonance is quantitatively described by a dimensionless number known as Q-factor or quality factor which is numerically equal to ratio of resonant frequency to bandwidth. The bandwidth is equal to L/R.

Note: The Q-factor of a series L-C-R circuit will be large if R is low, C is low or L is large.